A stick of length r units slides with its end | vedclass

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(D) Let the stick intercept the $x$-axis at point $(a, 0)$ and the $y$-axis at point $(0, b)$,and let $(x, y)$ be the midpoint of the stick. Then,$x =

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$\pi r$

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(D) Let the stick intercept the $x$-axis at point $(a, 0)$ and the $y$-axis at point $(0, b)$,and let $(x, y)$ be the midpoint of the stick. Then,$x = \frac{a+0}{2} \Rightarrow a = 2x$ and $y = \frac{0+b}{2} \Rightarrow b = 2y$. Given the length of the stick is $r$ units,we have $a^2 + b^2 = r^2$. Substituting the values of $a$ and $b$,we get $(2x)^2 + (2y)^2 = r^2$. $4x^2 + 4y^2 = r^2 \Rightarrow x^2 + y^2 = (\frac{r}{2})^2$. This represents a circle with center at the origin $(0, 0)$ and radius $R = \frac{r}{2}$. The length of the curve (circumference of the circle) is $2 \pi R = 2 \pi (\frac{r}{2}) = \pi r$.

$A$ stick of length $r$ units slides with its ends on coordinate axes. Then the locus of the midpoint of the stick is a curve whose length is

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